A106628 Anomalous prime numbers.

199, 211, 283, 317, 337, 389, 491, 509, 547, 577, 619, 683, 701, 773, 787, 797, 863, 887, 1069, 1109, 1129, 1153, 1163, 1373, 1381, 1409, 1459, 1523, 1531, 1571, 1627, 1637, 1669, 1709, 1723, 1733, 1759, 1831, 1889, 1913, 1933, 1951, 1979, 2003, 2017

Offset

1,1

Comments

If x and y are two consecutive prime numbers (x < y), Euclid's algorithm gives integers t and d such that tx+dy = 1 = gcd(x, y). The algorithm "Anomalia" gives t and d such that |t+d| is as small as possible (it is often = 1). The prime number x is 'anomalous' iff |t+d| > 1 for x and y.

That is, primes p such that neither q-1 nor q+1 is divisible by q-p, where q is the next prime larger than p. - Charles R Greathouse IV, Aug 20 2017

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

Conjecture: a(n) ~ n log n. - Charles R Greathouse IV, Aug 20 2017

Example

a(1) = 199 because -88*199+83*211 = 1, |-88+83| = 5 > 1;
|tx+dy| = 1 for all primes x < 199 (when t and d are determined by the algorithm "Anomalia")

Prog

{ALGORITHM "Anomalia" in pseudo language} INPUT x, y {positive integers} m := x, n := y, b := 0, d := 1, p := 1, t := 0 WHILE m <> 0 DO q := n DIV m h := m, m := n-q*m, n := h h := b, b := d-q*b, d := h h := p, p := t-q*p, t := h WRITE - The gcd of the numbers is WRITE n = tx+dy {this is 1 for consecutive primes}
(PARI) is(x)=if(!isprime(x), return(0)); my(y=nextprime(x+1), d=y-x); (y-1)%d && (y+1)%d \\ Charles R Greathouse IV, Aug 20 2017

Crossrefs

Subsequence of A083371.

Sequence in context: A171391 A151738 A128312 * A109561 A031930 A160342

Adjacent sequences: A106625 A106626 A106627 * A106629 A106630 A106631

Keyword

base,nonn

Author

Pahikkala Jussi, May 11 2005

Status

approved